Winter school « Dynamics and Pde » at Saint-Etienne de Tinée, 2017

Mini-courses:

Abstract : Consider the Schr\"odinger equation
$$\ir \dot\psi=-\Delta\psi+V(x)\psi+\epsilon W(x,-\im\nabla,\omega t)\psi\ ,\quad x\in R^d$$
with $V$ a potential growing at infinity at list
quadratically and $W$ a perturbation which depends
quasiperiodically on time.

I will present a method based on pseudodifferential calculus and KAM
theory in order to prove reducibility of the system, namely existence
of a unitary transformation, quasiperiodically dependent on time,
which conjugates the system to a time independent one. Boundedness of
Sobolev norms and pure point nature of the Floquet spectrum
follow. A striking connection between classical and quantum
perturbation theory appears.

In the 1-dimensional case the result applies to quite general situations
including magnetic perturbations and also to perturbations growing at
infinity as fast as the unperturbed potential $V$.

In the $d$-dimensional case reducibility is proved in the case
where
$$V(x)=\sum_{j=1}^d\nu_j^2x_j^2\ ,\quad \nu_j>0$$ with $W$ an
operator with symbol which is a polynomyal of degree 2 in $x$ and
$\xi$ with coefficients quasiperiodic in time. The result in the case
$d>1$ has been obtained in collaboration with B. Gr\'ebert, A. Maspero
and D. Robert.

Abtract: The goal of this mini course is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension,

with periodic, even in space, initial data of small size , is almost globally defined in time on Sobolev spaces,  as soon as the initial data are smooth enough, and

the gravity-capillarity parameters are taken outside an exceptional subset of zero measure.  Since the water waves equations are a quasi-linear system, usual normal forms approaches would face the well known problem of losses of derivatives in the unbounded transformations. To overcome such a difficulty, after a paralinearization of the capillarity-gravity water waves equations, necessary to obtain energy estimates, and thus local existence of the solutions, we first perform several paradifferential reductions of the

equations to obtain a diagonal system with constant coefficients symbols, up to smoothing remainders. Then we may start with a normal form procedure

where the small divisors are compensated by the previous paradifferential regularization. The reversibility structure of the water waves equations, and

the fact that we look for solutions even in x, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions. This is a joint work with J.M. Delort.

*T. Hmidi : "Relative  equilibria in geophysical flows"

Abstract  We shall discuss the existence of rotating and translating vortex  patches for various nonlinear transport equations arising in fluid dynamics like  Euler   and  quasi-geostrophic equations.    The material support of this mini-course is several  joint works with de la Hoz, Hassainia, Mateu and Verdera.

F. Planchon : "On the growth of Sobolev norms for NLS on compact manifolds"

Abstract :We review recent joint work with Nicola Visciglia and N. Tzvetkov, revisiting the issue of upper bounds on the time evolution of higher order Sobolev norms for nonlinear Schrodinger equations on compact manifolds. We develop an elementary method to recover (and sometimes extend) previous results, all of which were, up to now, connected to seminal work of Bourgain and relying on rather sophisticated technologies.

Abstract : We review recent joint work with Nicola Visciglia and N. Tzvetkov, revisiting the issue of upper bounds on the time evolution of higher order Sobolev norms for nonlinear Schrodinger equations on compact manifolds. We develop an elementary method to recover (and sometimes extend) previous results, all of which were, up to now, connected to seminal work of Bourgain and relying on rather sophisticated technologies.

Talks :

Abstract: This talk regards the  following question for the weak Leray-Hopf solutions of the three dimensional  Navier-Stokes equations. Namely, what are sufficient classes of initial data that ensure uniqueness of them on some time interval?

*N. Burq: "Quasi-invariant measures for the 1-d NLS".

*L. Chierchia : "The steep Nekhoroshev's theorem "

Abstract: The talk will concern a recent version of Nekhoroshev's Theorem on the long-times stability of action variables for nearly-integrable, analytic Hamiltonian systems [Guzzo, Chierchia, Benettin, CMP 342, 2016]. In particular, I will discuss the dependence upon the so-called steepness indices of the stability exponents, which we conjecture to be optimal.

*E. Haus : "KAM for beating solutions of the quintic NLS"

Abstract: We consider the nonlinear Schrödinger equation of degree five on the circle. We prove the existence of quasi-periodic solutions which bifurcate from “resonant” solutions (already studied by Benoît Grébert and Laurent Thomann) of the system obtained by truncating the Hamiltonian after one step of Birkhoff normal form, exhibiting recurrent exchange of energy between some Fourier modes. The existence of these quasi-periodic solutions is a purely nonlinear effect. This is a joint work with Michela Procesi.

*G. Iooss: "Existence of bifurcating patterns in steady Bénard Rayleigh convection"

Abstract:  This is a joint work with B.Braaksma. Extending the results obtained in the sixties for bifurcating periodic patterns, by Yudovich et al, Kirchgässner et al, Rabinowitz…., existence of bifurcating quasipatterns in the steady Bénard-Rayleigh convection problem is proved. These are two-dimensional patterns, quasiperiodic in any horizontal direction, invariant under horizontal rotations of angle π/q. There is a small divisor problem for q≥4.

Using the results of Berti-Bolle-Procesi, developped by the author with Braaksma and Stolovitch on the Swift-Hohenberg PDE model, we adapt it to a Navier-Stokes system ruling the Bénard-Rayleigh convection problem.

Our solution is approximated by the truncated power series which is formally obtained, but which is divergent in general (Gevrey series).

First, we formulate the problem in introducing a new parameter, using the freedom we have for the parametrization of bifurcating solutions, as we did on the Swift-Hohenberg PDE model. For using the Nash-Moser process, we are faced with the problem of inverting  a linear operator which is the differential at a non zero point.

There are two new difficulties:

I) first, the extra dimension leading to a more complicated spectrum of the linear operator. This first difficulty leads to use specific projections for reducing the spectrum of the studied operator L, which we want to invert, to a finite set very close to 0.

ii) The second difficulty is the fact that the linearization at a non zero point leads to a non selfadjoint operator, contrary to what occurs in previous works. This is more serious, and leads to use the spectrum of LL* which depends mainly quadratically on the parameter.

A careful study of the « bad set » of parameters allows to obtain a good estimate, as it is necessary for using the results of Berti-Bolle - Procesi for solving « the range equation ». Note that, contrary to previous works mentioned above,  it becomes useless here to use the Bourgain results for obtaining an estimate in high Sobolev norms.

It then remains to solve the one-dimensional "bifurcation equation ».

For any q≥4 and provided that some coefficient is not zero,  and a transversality condition is realized, we prove the existence of a bifurcating convective quasipattern of order 2q, above the critical Rayleigh number.

Abstract： In this talk, I will firstly introduce some basic concepts about ( not necessarily Hamiltonian) integrable systems. Then I will present a geometric linearization theorem proved by N.T.Zung and also a conjecture of him. I will apply a theorem of Marc Chaperon about smooth conjugacy of weakly hyperbolic systems to partially answer Zung's conjecture. At last, I will discuss the geometric linearization for some other integrable systems. I will give more results on the conjecture under certain additional conditions.

Abstract: In this talk we consider time dependent Schr\"odinger linear PDEs of
the form $\im \partial_t \psi = (H +V(t)) \psi$ where $V(t)$ is a
perturbation smooth in time and $H$ is a self-adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing. We prove
that the Sobolev norms of the solution grow at most as $t^\epsilon$ when $t\mapsto \infty$, for any $\epsilon >0$.
If $V(t)$ is analytic in time we improve the bound to $(\log t)^\gamma$, for some $\gamma >0$. The proof follows the strategy of adiabatic approximation of the flow. We recover most of known results and obtain new estimates for several models including $1$-degree of freedom Schr\"odinger operators on $\R$. This is a joint work with Didier Robert.

*I. Naumkin :"Scattering for the NLS equation."

Abstract: "In this talk we briefly discuss some regulatory issues that appear in the NLS equation and their relation to the scattering problem for this equation. Then, under some assumptions, we prove the existence of scattering for the NLS equation with any scattering-supercritical nonlinearity. In the case of the NLS equation with scattering-critical nonlinearities, we prove the existence of modified-scattering and we present the long-time asymptotics for the solutions to the NLS equation."

Abstract : "This talk is devoted to the homogenization of elliptic equations with oscillating (periodic) coefficients and oscillating (periodic) boundary data. We will explain recent quantitative results in collaboration with Armstrong, Kuusi and Mourrat. We will focus on the study of the resonances created by the interaction between the geometry of the domain and the underlying periodic lattice."

*M. Procesi: "Long-time stability of small finite gap solutions of the  cubic NLS on T^2"

Abstract: I will discuss a recent result in collaboration with Alberto Maspero, concerning the stability of finite gap solutions of the NLS on T^2.  More precisely we study the 2d NLS close to  a "generic" finite gap solution of the 1d NLS and show that we may perform two steps of Birkhoff normal form.